Short answers, pulled from the story.
The four disputed islands are Iturup (Etorofu), Kunashir (Kunashiri), Shikotan, and the Habomai Islands. They form the southernmost part of the Kuril chain, which stretches between Hokkaido, Japan, and Russia's Kamchatka Peninsula. Japan calls them the Northern Territories and claims they are part of Hokkaido Prefecture.
The core obstacle is the territorial dispute over the four southernmost Kuril Islands, which the Soviet Union seized in August-September 1945. The Soviet-Japanese Joint Declaration of 1956 ended the formal state of war but did not produce a peace treaty, as the two sides could not agree on which islands would be returned. That disagreement has persisted, and Russia's withdrawal from peace treaty talks in March 2022, following Japan's sanctions over the invasion of Ukraine, has stalled negotiations further.
The declaration, signed on the 19th of October 1956 in Moscow, stipulated that the Soviet Union would hand over the Habomai Islands and Shikotan to Japan, but only after a formal peace treaty was concluded. The two larger islands, Iturup and Kunashiri, were not addressed. Japan interpreted the declaration as a starting point for further negotiations, while the Soviet Union considered the territorial question resolved.
Article 2c of the treaty, signed on the 8th of September 1951, states that Japan renounces all right, title, and claim to the Kurile Islands. The Soviet Union refused to sign the treaty. The treaty did not explicitly assign sovereignty over the islands to the Soviet Union, and the U.S. State Department later clarified that the Habomai Islands and Shikotan are properly part of Hokkaido and that Japan is entitled to sovereignty over them.
Russia maintains that all the Kuril Islands are legally part of Russia as a result of World War II, citing the explicit language of the Yalta Agreement. In March 2022, Dmitry Medvedev stated that negotiations had always been "ritualistic" and that Russia's 2020 constitutional amendments, which bar the alienation of Russian territory, had closed the question. A July 2009 poll found 89% of Russian respondents opposed any territorial concessions to Japan.
Following Russia's invasion of Ukraine in 2022, Japan imposed sanctions on Russia and reverted to harder-line language, with Prime Minister Fumio Kishida declaring on the 7th of March 2022 that the southern Kurils are a territory over which Japan has sovereignty. Russia responded on the 21st of March 2022 by withdrawing from peace treaty talks and freezing joint economic projects. Russia subsequently withdrew from a visa agreement allowing former Japanese residents to visit the islands.